2.1. Dynamics and Attitude determination 23
2.1.2. Attitude dynamics 32
51.5354s, or 100.71473◦
Figure 2.6: Earth‐Centered Earth‐Fixed Reference Frame (obtained from reference 14)
Equations (2.1) – (2.19) were obtained from reference 14
2.1.2 Attitude Dynamics
Attitude dynamics describe the orientation of a body in an orbit and can be explained using
rotations. When examining attitude dynamics, it is important to describe the reference frames
being used to give a basis for the rotations.
Reference Frames
Three main reference frames are used to describe the orientation, or attitude, of a spacecraft in
orbit. These are the inertial, orbital, and body frames.
Inertial frame:
An inertial frame is used for attitude applications. The X direction points from the focus of the orbit
to the vernal equinox, Υ, the Z direction is in the orbital angular velocity direction, and Y is
Orbital frame:
The orbital frame is located at the mass center of the spacecraft, and the motion of the frame
depends on the orbit. This frame is non‐inertial because of orbital acceleration and the rotation of
the frame. The ‘ˆ3’ axis is in the direction from the spacecraft to the Earth, ‘ˆ2’ is the direction
opposite to the orbit normal, and ‘ˆ1’ is perpendicular to ‘ˆ2’ and ‘ˆ3’. In circular orbits, ‘ˆ1’ is the
direction of the spacecraft velocity. The three directions ‘ˆ1’, ‘ˆ2’, and ‘ˆ3’ are also known as the roll,
pitch, and yaw axes, respectively. Figure 2.7, shows a comparison of the inertial and orbital frames
in an equatorial orbit.
Figure 2.7: Earth‐Centered Inertial and Orbital Reference Frames (obtained from reference 14)
Body Frame:
Like the orbital frame, the body frame has its origin at the spacecraft’s mass center. This frame is
fixed in the body, and therefore is non‐inertial. The relative orientation between the orbital and
body frames is the basis of attitude dynamics and control.
Principal Axis:
Like the orbital frame, the body frame has its origin at the spacecraft’s mass center. This frame is
fixed in the body, and therefore is non‐inertial. The relative orientation between the orbital and
body frames is the basis of attitude dynamics and control.
Rotations
Rotations and transformations are performed to obtain the desired vector in alternate reference
frames. Two notations commonly used to describe this rotation are Euler angles and quaternions.
Rotations are discussed by Hall. Obtained from reference 16
Rotation Matrix:
The relationship between vectors expressed in different reference frames is described as:
(2.20)
Where
′
′
is the rotation from ‘Fb’ to ‘Fi’,’vi’ is a vector in ‘Fi’, and ‘vb’ is the same vector in ‘Fb’. The components of a rotation matrix are the direction cosines of the two sets of reference axes. Ingeneral,
(2.21)
Where ‘ ′ is the cosine of the angle between the x axis of the first frame and the x axis of the second frame. The rotation matrix from the inertial reference frame to the orbital reference
frame is defined
(2.22)
Which is a
rotation, based on the orbital elements.
Quaternions:
A second way to express a rotation is through the use of quaternions. The quaternion set, ′ ′, is a 4
(2.23)
The rotation matrix in terms of the quaternion is:
(2.24) 1 2 2
Where ′ ′ is the skew symmetric of ′ ′ defined as:
(2.25)
In addition, ′ ′can be expressed in terms of ‘R’ as:
(2.26)
Quaternions have advantages and disadvantages over rotation matrix notation. The singularities
that exist when certain Euler angles are small are eliminated with the use of quaternions. However,
the physical meaning of quaternions is obscure and not as intuitive as rotation angles.
Angular velocity:
The angular velocity, ‘ω’, is used to examine the angular displacements that occur over time.
Angular velocities are dependent on the frame of reference, and are designated by ′ ′, which is
a rotation of ‘Fc’ with respect to ‘Fa’ as seen by ‘Fb’
Angular velocities add, but only when they are in the same reference frame. For example, the
following relation is valid:
(2.27)
When the angular velocities are in different reference frames, however, it is necessary to perform
rotations. This is evident in:
(2.28)
Where the angular velocity is seen by the body frame. Equation of motion
Equation if motion, is a derivation of equations of motion for a satellite system.
Dynamic equation of motion:
The Rotational equations for a rigid body are derived by beginning with the rotational equivalent
of: (2.29)
Or (2.30)
Where ′ ’ the angular momentum about the mass center, and ‘ ’ is the torque. This relationship is
represented in matrix form by:
(2.31)
Assuming that the body frame is fixed to the body at the mass center, the angular momentum can
be represented by:
(2.32)
Which leads to:
(2.33)
Solving for leads to:
(2.34)
In this study, the only torques,’ are gravity‐gradient and magnetic is expanded into: (2.35)
If principle axes are used, these equations are known as Euler’s Equations.
Kinematic Equations of motion:
The kinematic equations of motion are obtained by beginning with the definition of a quaternion.
The quaternion is in the form:
(2.36)
is equal to: (2.37)
Since ′∆ ′ is infinitesimal and ′∆Φ ∆t′, where ‘ω’ is the magnitude of the instantaneous
angular velocity of the body, the following small angle assumptions are used:
(2.38)
This leads to:
(2.39)
Where ‘Ω’ is the skew symmetric matrix: (2.40)
The derivative of the quaternion is:
(2.41)
(2.42)
This equation represents the kinematic equation of motion of the spacecraft.
Equations (2.20) – (2.42) obtained from reference 16
2.1.3 Attitude determination
The orientation of a spacecraft can be determined by describing the rotation between a spacecraft
fixed reference frame and a known reference frame. This description is accomplished by finding
rotations between measured attitude vectors and known quantities. For example, a Sun sensor
determines the vector from the spacecraft to the Sun in the body frame, ‘sb’. Since the vector in the inertial frame, ‘si’ , can be calculated from ephemeris data, the following relation is useful:
(2.43)
The attitude is determined by solving for ‘Rbi’. This equation does not have a unique solution,
however, so it is necessary to obtain a second attitude measurement to fully describe the attitude
of a spacecraft.
Common attitude sensors include Sun sensors, Earth sensors, magnetometers, star trackers, and
gyroscopes, and are described by Wertz and Larson. Obtained from reference 17 A comparison of the ranges of these attitude sensors is shown in Figure 2.9: